## Abstract

Let n ∈ ℕ\{0, 1}. Let q be the n × n diagonal matrix with entries q_{11},..., q_{nn} in] 0, +∞[. Then qℤ^{n} is a q-periodic lattice in ℝ^{n} with fundamental cell Q ≡ Π^{n} _{j=0}]0, q_{jj}[. Let p ∈ Q. Let Ω be a bounded open subset of ℝ^{n} containing 0. Let G be a (nonlinear) map from ∂Ω × ℝ to ℝ. Let γ be a positive-valued function defined on a right neighbourhood of 0 in the real line. Then we consider the problem for ε > 0 small, where ν_{p+εΩ} denotes the outward unit normal to p + ε∂Ω. Under suitable assumptions and under condition lim_{ε→0+}γ(ε)^{-1}ε ∈ ℝ, we prove that the above problem has a family of solutions {u(ε, ·)}_{ε∈]0, ε′[} for ε′ sufficiently small, and we analyse the behaviour of such a family as ε approaches 0 by an approach which is alternative to those of asymptotic analysis.

Original language | English |
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Pages (from-to) | 511-536 |

Number of pages | 26 |

Journal | Complex Variables and Elliptic Equations |

Volume | 58 |

Issue number | 4 |

Early online date | 10 Jan 2012 |

DOIs | |

Publication status | Published - 01 Apr 2013 |

## Keywords

- Laplace operator
- periodic nonlinear Robin boundary-value problem
- real-analytic continuation in Banach space
- singularly perturbed data
- singularly perturbed domain